As suggested in a comment to my previous post, pentagons will make a sphere, or a ball, when they are sewed together.
It has already been done by lots of people, so here are a few examples, and also tutorials:
Jinny Beyer wrote a whole book about making patchwork balls, using both pentagons and other shapes. There was not much information on her website about the balls, so here is a link to
where there are many more photos of different balls from the book.
Here is an example of an even more interesting ball, construction wise.
Interesting challenge, indeed!
I might just have to get that book.